Ten lectures on wavelets
Wavelets based on orthogonal polynomials
Mathematics of Computation
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In this paper we present a recursive approach to generate complex orthogonal polynomial systems. The systems belong to a class of polynomial wavelets successfully introduced by Skopina [M. Skopina, Orthogonal polynomial Shauder bases in C[-1,1] with optimal growth of degrees, Sb. Math. 192 (3) (2001) 433-454; M. Skopina, Multiresolution analysis of periodic functions, East J. Approx. 3 (1997) 203-224]. Consequently, by using the obtained recursive-type relation, it is possible to generate a great variety of complex polynomial functions which satisfy useful wavelet-like properties. We prove some additional multiscale results concerning these systems. More precisely, we state a practical two-scale relation and the decomposition and reconstruction formulae which determine the multiresolution analysis framework. From the reconstruction formula, we obtain the recursive approach which provides the Skopina's systems. Finally, a numerical example in which explicit complex orthogonal polynomials are found recursively is presented.